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  <title>机械设计基础——平面机构的自由度和速度分析 - Danaの小岛</title>

  
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<widget class="widget-wrapper toc single" id="data-toc"><div class="widget-header cap dis-select"><span class="name">机械设计基础——平面机构的自由度和速度分析</span></div><div class="widget-body fs14"><div class="doc-tree active"><ol class="toc"><li class="toc-item toc-level-2"><a class="toc-link" href="#%E7%BB%AA%E8%AE%BA%E6%A6%82%E5%BF%B5%E8%BE%A8%E6%9E%90"><span class="toc-text"> 绪论概念辨析:</span></a></li><li class="toc-item toc-level-2"><a class="toc-link" href="#%E8%BF%90%E5%8A%A8%E5%89%AF%E5%8F%8A%E5%85%B6%E5%88%86%E7%B1%BB"><span class="toc-text"> 运动副及其分类</span></a></li><li class="toc-item toc-level-2"><a class="toc-link" href="#%E6%9C%BA%E6%9E%84%E8%BF%90%E5%8A%A8%E7%AE%80%E5%9B%BE%E7%9A%84%E7%BB%98%E5%88%B6"><span class="toc-text"> 机构运动简图的绘制</span></a><ol class="toc-child"><li class="toc-item toc-level-3"><a class="toc-link" href="#%E7%BB%98%E5%88%B6%E5%9B%BE%E7%A4%BA%E4%B8%8E%E6%B3%A8%E6%84%8F"><span class="toc-text"> 绘制图示与注意</span></a></li><li class="toc-item toc-level-3"><a class="toc-link" href="#%E7%BB%98%E5%88%B6%E6%AD%A5%E9%AA%A4"><span class="toc-text"> 绘制步骤</span></a></li></ol></li><li class="toc-item toc-level-2"><a class="toc-link" href="#%E5%B9%B3%E9%9D%A2%E6%9C%BA%E6%9E%84%E7%9A%84%E8%87%AA%E7%94%B1%E5%BA%A6"><span class="toc-text"> 平面机构的自由度</span></a></li><li class="toc-item toc-level-2"><a class="toc-link" href="#%E9%80%9F%E5%BA%A6%E7%9A%84%E8%AE%A1%E7%AE%97-%E9%80%9F%E5%BA%A6%E7%9E%AC%E5%BF%83%E6%B3%95"><span class="toc-text"> 速度的计算 (速度瞬心法)</span></a><ol class="toc-child"><li class="toc-item toc-level-3"><a class="toc-link" href="#%E7%9E%AC%E5%BF%83%E7%9A%84%E7%A1%AE%E5%AE%9A"><span class="toc-text"> 瞬心的确定</span></a><ol class="toc-child"><li class="toc-item toc-level-4"><a class="toc-link" href="#%E7%9B%B8%E8%BF%9E%E6%9E%84%E4%BB%B6%E7%9A%84%E7%9E%AC%E5%BF%83"><span class="toc-text"> 相连构件的瞬心</span></a></li><li class="toc-item toc-level-4"><a class="toc-link" href="#%E4%B8%89%E5%BF%83%E5%AE%9A%E7%90%86"><span class="toc-text"> 三心定理</span></a></li></ol></li><li class="toc-item toc-level-3"><a class="toc-link" href="#%E9%80%9F%E5%BA%A6%E7%9A%84%E6%B1%82%E8%A7%A3"><span class="toc-text"> 速度的求解</span></a></li></ol></li></ol></div></div></widget>




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<article class='md-text content post reveal'>
<h1 class="article-title"><span>机械设计基础——平面机构的自由度和速度分析</span></h1>
<h2 id="绪论概念辨析"><a class="markdownIt-Anchor" href="#绪论概念辨析"></a> 绪论概念辨析:</h2>
<ul>
<li>构件: 独立的运动单元</li>
<li>零件: 独立的制造单元</li>
<li>机构: 能够用来传递运动和力或改变运动形式的<mark>构件</mark>组合体</li>
<li>机器: 根据某种使用要求而设计的<mark>构件</mark>的组合体</li>
</ul>
<h2 id="运动副及其分类"><a class="markdownIt-Anchor" href="#运动副及其分类"></a> 运动副及其分类</h2>
<p>两构件组成运动副, 其运动副元素不外乎点、线、面。按照接触特性通常把运动副分成以下几类：</p>
<ol>
<li>
<p>低副——面接触，应力低</p>
<ul>
<li>转动副：若组成运动副的两构件只能在平面内相互转动，这种运动副称为转动副</li>
<li>移动副：若组成运动副的两构件只能沿某一方向做相对对直线移动，这种运动副称为移动副</li>
</ul>
</li>
<li>
<p>高副——点、线接触，应力高<br />
两构件通过点或线接触组成的运动副称为高副</p>
</li>
</ol>
<h2 id="机构运动简图的绘制"><a class="markdownIt-Anchor" href="#机构运动简图的绘制"></a> 机构运动简图的绘制</h2>
<h3 id="绘制图示与注意"><a class="markdownIt-Anchor" href="#绘制图示与注意"></a> 绘制图示与注意</h3>
<p><img class="lazy" src="" data-src="https://img.gejiba.com/images/c6fbe924e9f17f202fd697a55a03a205.png" alt="运动简图绘制" /><br />
<img class="lazy" src="" data-src="https://img.gejiba.com/images/43074dbd3e210fae23092c6dac8f33aa.png" alt="机械简图绘制2" /><br />
在机械简图中，<mark>数字</mark> 表示构件，<mark>字母</mark> 表示运动副</p>
<h3 id="绘制步骤"><a class="markdownIt-Anchor" href="#绘制步骤"></a> 绘制步骤</h3>
<p>机构中的构件可分为以下三类：</p>
<ol>
<li>固定构件（机架）：用来支承活动构件的构件</li>
<li>原动件（主动件）：运动规律已知的活动构件</li>
<li>从动件：机构中随原动件运动而运动的其他构件</li>
</ol>
<p>画构件时应当撇开构件的实际外形，只考虑运动副的性质。</p>
<ol>
<li>确定构件数目</li>
<li>确定运动副的种类和数目</li>
<li>选定适当的比例尺</li>
<li>将机架画上阴影线，并在原动件注明运动方向</li>
</ol>
<h2 id="平面机构的自由度"><a class="markdownIt-Anchor" href="#平面机构的自由度"></a> 平面机构的自由度</h2>
<p>平面机构由 n 个活动构件构成（即除去机架），若机构中低副数量为 P<sub>L</sub>个, 高副个数为 P<sub>H</sub>个, 自由度为 F, 则</p>
<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>F</mi><mo>=</mo><mn>3</mn><mi>n</mi><mo>−</mo><mn>2</mn><msub><mi>P</mi><mi>L</mi></msub><mo>−</mo><msub><mi>P</mi><mi>H</mi></msub></mrow><annotation encoding="application/x-tex">F = 3n -2P_L -P_H
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.13889em;">F</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.72777em;vertical-align:-0.08333em;"></span><span class="mord">3</span><span class="mord mathdefault">n</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord">2</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">L</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.08125em;">H</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></p>
<p>其中应注意有复合铰链、虚约束、局部自由度的情况。</p>
<ul>
<li>复合铰链下具有（N-1）个低副</li>
<li>对于那些不对构件运动造成影响的约束为虚约束, 应当忽视</li>
<li>对于可旋转滚子, 其转动副不影响确定运动, 却引入了一个自由度, 应去除这种局部自由度</li>
<li>对于对称结构 (如行星轮系)应只计算一个独立部分<br />
<img class="lazy" src="" data-src="https://img.gejiba.com/images/9e980900a2bfe6868d55c71d666ee863.png" alt="特例" /></li>
</ul>
<h2 id="速度的计算-速度瞬心法"><a class="markdownIt-Anchor" href="#速度的计算-速度瞬心法"></a> 速度的计算 (速度瞬心法)</h2>
<h3 id="瞬心的确定"><a class="markdownIt-Anchor" href="#瞬心的确定"></a> 瞬心的确定</h3>
<h4 id="相连构件的瞬心"><a class="markdownIt-Anchor" href="#相连构件的瞬心"></a> 相连构件的瞬心</h4>
<p>在任一瞬时, 两构件间的相对运动可看做绕某一重合点的转动, 该重合点称为速度瞬心或瞬时回转中心, 简称<mark>瞬心</mark>。<br />
如果这两个构件都在运动则称其瞬心为<strong>相对瞬心</strong>，如果两构件之一是静止的则称<strong>绝对瞬心</strong>。</p>
<ul>
<li><mark>所有机架与构件的瞬心均为绝对瞬心</mark></li>
</ul>
<p>瞬心的确定有以下几种情况：</p>
<ol>
<li>两构件组成<strong>转动副</strong>时，<em>转动副中心</em>为其瞬心</li>
<li>两构件组成<strong>移动副</strong>时，其瞬心位于<em>移动导路垂线的无穷远处</em></li>
<li>两构件组成<strong>纯滚动高副</strong>时，<em>接触点</em>就是其瞬心</li>
<li>两构件组成<strong>滑动兼滚动的高副</strong>时，其瞬心位于<em>过接触点的公法线 n-n 上</em>，具体位置需要其他条件确定<br />
<img class="lazy" src="" data-src="https://img.gejiba.com/images/b30a59330e5a006ea34f0d11d18356b9.png" alt="速度瞬心的确定" /></li>
</ol>
<h4 id="三心定理"><a class="markdownIt-Anchor" href="#三心定理"></a> 三心定理</h4>
<p>对于不直接接触的各个构件, 其瞬心可用<mark>三心定理</mark>寻求。即做相对平面运动的三个构件共有三个瞬心，这三个瞬心位于同一直线上</p>
<ul>
<li>各构件间的联系可用画多边形的方法寻找，如：<br />
若有一四杆机构标号为 1,2,3,4 首尾相连，则可抽象成一个正方形进行三心定理的求解<br />
<img class="lazy" src="" data-src="https://img.gejiba.com/images/022afb1d2b9afb7dd1f7bdd4345de4e9.png" alt="速度瞬心法" /></li>
</ul>
<h3 id="速度的求解"><a class="markdownIt-Anchor" href="#速度的求解"></a> 速度的求解</h3>
<p>两构件的角速度与其绝对瞬心至相对瞬心的距离成反比。<br />
设机架为 1 号构件，则</p>
<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mfrac><msub><mi>ω</mi><mi>i</mi></msub><msub><mi>ω</mi><mi>j</mi></msub></mfrac><mo>=</mo><mfrac><mrow><msub><mi>P</mi><mrow><mn>1</mn><mi>j</mi></mrow></msub><msub><mi>P</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow><mrow><msub><mi>P</mi><mrow><mn>1</mn><mi>i</mi></mrow></msub><msub><mi>P</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{\omega_i} {\omega_j} = \frac{P_{1j} P_{ij}} {P_{1i} P_{ij}} 
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.079668em;vertical-align:-0.972108em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1075599999999999em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">ω</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">ω</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.972108em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.332438em;vertical-align:-0.972108em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.36033em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mathdefault mtight">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.972108em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p>
<p>若求线速度只需使用</p>
<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>v</mi><mo>=</mo><mi>ω</mi><mi>r</mi></mrow><annotation encoding="application/x-tex">v = \omega r
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">ω</span><span class="mord mathdefault" style="margin-right:0.02778em;">r</span></span></span></span></span></p>
<p>其中, r 为求速度点到速度瞬心的距离。</p>



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